examday

/TMUA

For the sequence $\{a_n\}$ ($a_n\in\N^+$), we define $b_k$ as the maximum in $a_1,a_2,\ldots,a_k$ ($k=1,2,\ldots n$) ($n\in\N^+$). We call the sequence $\{b_n\}$ the “M sequence” of $\{a_n\}$.

Which of the following statements are necessarily true?

• I. If $\{a_n\}$ is a monotonically decreasing sequence, then $\{b_n\}$ is a constant sequence.
• II. If $\{a_n\}$ is a monotonically increasing sequence, then $b_n=a_n$.
• III. There are 8 forms of $\{a_n\}$ that can satisfy $\{b_n\}=2,3,3,5,5$.